We establish the Curry-Howard isomorphism between constructive classical logic and CPS-calculus. CPS-calculus exactly means the target language of Continuation Passing Style(CPS) transforms. Constructive classical logic we refer to are LKT and LKQ introduced by Danos et al.(1993). Keywords: Constructive Classical Logic,CPS-calculus,CPS-transform,CPS-semantics 1. Introduction 1.1. What is Constructive Classical Logic? It has long been thought that classical logic cannot be put to use for computational purposes. It is because, in general, the normalization process for the the proof of classical logic has a lot of critical pairs. Classical logic we consider in this paper is Gentzen's sequent-style classical logic (i.e., LK) and its variants. In this context, above fact is related to the non-deterministic behavior of cut-elimination. Of course, by Gentzen's theorem, LK has a Strongly Normalizable(SN) cut-elimination procedure. The problem is, it is not Church-Rosser(CR). Constructive ...

The minimal relevant logic B+, seen as a type discipline, includes an extension of Curry types known as the intersection type discipline. We will show that the full logic B+ gives a type assignment system which produces a model of Plotkin’s call-by-value-calculus. 1

Type free lazy -calculus is enriched with angelic parallelism and demonic nondeterminism. Call-by-name and call-by-value abstractions are considered and the operational semantics is stated in terms of a must convergence predicate. We introduce a type assignment system with intersection and union types and we prove that the induced logical semantics is fully abstract.

. To model at the same time parallel and non-deterministic functional calculi we define a powerdomain functor P such that it is an endofunctor over the category of algebraic lattices. P is locally continuous and we study the initial solution D 1 of the domain equation D = P([D ! D]? ). We derive from the algebras of P the logic of D 1 , that is the axiomatic description of its compact elements. We then define a -calculus and a type assignment system using the logic of D 1 as the related type theory. We prove that the filter model of this calculus, which is isomorphic to D 1 , is fully abstract with respect to the observational pre-order of the -calculus. Keywords: -calculus, Non-determinism, Full Abstraction, Powerdomain Construction, Intersection Type Disciplines. 1. Introduction One of the main issues in the design of programming languages is the achievement of a good compromise between the multiplicity of control structures and data types and the unicity of the mathematica...

INTRODUCTION Lambda calculus, invented by Alonzo Church in the 1930s, is a general but syntactically simple model of computation. It was conceived as part of a system of higher-order logic and function theory. The first undecidability results were for lambda calculus; similar results for Turing machines came later. In addition to its purely mathematical applications, lambda calculus is important in the study of computer programming languages. It has served as a basic linguistic prototype from which LISP, ALGOL-like languages, and functional programming languages have been derived. It also serves as a basic metalanguage for expressing the denotational semantics of programming languages. These notes are a brief introduction to the type-free lambda calculus. Two versions of the type-free lambda calculus are presented: the call-by-name (CBN) and the call-by-value (CBV) calculi. The CBN calculus was the original version of the lambda

This paper is an introduction to intersection type disciplines, with the aim of illustrating their theoretical relevance in the foundations of λ-calculus. We start by describing the well-known results showing the deep connection between intersection type systems and normalization properties, i.e., their power of naturally characterizing solvable, normalizing, and strongly normalizing pure λ-terms. We then explain the importance of intersection types for the semantics of λ-calculus, through the construction of filter models and the representation of algebraic lattices. We end with an original result that shows how intersection types also allow to naturally characterize tree representations of unfoldings of λ-terms (Böhm trees).